Popula spe ifi atio
• If an agent has accepted an alternative with v, her value is
= + � + − ,
while the value of searching is
= − + max� � න
0
� I + − � �
Keisuke Kawata
ISS, UTokyo
Search decision making
• The last slide characterized the optimal search strategy with an exogenous distribution of alternative values.
←We cannot discuss the behavior response of search strategy and the value distribution with their interactions.
Plan of talk
1. Two firms example 2. Infinitely many firms 3. Discussion
Key concepts: Endogenous value distribution, price posting, Diamond paradox
Price posting model: Sellers post (and commit) price before search activity.
← Straightforward way of endogenous value distribution
• Let start from a simple labor market model where 2 firms (j=A,B) and L workers.
• Fi ’s pa off is � = � − � , where y is the marginal revenue (exogenous), is the wage, and � is the number of hired workers.
• Wo ke i’s pa off is − � where c is search costs (exogenous), is accepted wage, and � is the number of visited firms.
> If a worker accepts no job offers, her payoff is 0.
• Timing of game
1. Firms simultaneously post wage .
2. Workers engage the sequential search activity.
Structure of information: Workers cannot know posted wage of each firm before visiting them, alternatively, they have a belief of the wage distribution.
Equilibrium concepts: Sub-game perfect equilibrium with Bayesian belief.
• Let focus on the symmetric equilibrium where both firms offer same wages. Equilibrium is defined by posted wage �, optimal search strategy, and consisted belief.
• Let check the non-deviated symmetric posting wage.
• Let suppose a firm A deviates the equilibrium wage �.
• In the second stage,
Wo ke s elief of the age dist i utio i the e t fi is; Pr = � = , and Pr ≠ � = .
Equilibrium search activity is as follows.
• When a worker firstly visits the firm B, she must accept the job offer.
• When a worker firstly visits the firm A, she accepts the offer if and only if
� > � − .
• Fi ’s pa off is � − � ��, where
�� = if � < � − (or � < ),
�� = �2 if � ≥ � − (and � ≥ ).
⇒ The symmetric equilibrium holds where � =0. (monopolistic wage). Even if the search cost c is very small, above equilibrium holds.
←Diamond paradox (Diamond 1974).
Intuition: Bertrand paradox
• If c is exactly equal to zero, the model is equivalent to the Bertrand model.
⇒ Fi ’s pa off is � − � ��, where
�� = � if � > � and � ≥ ,
�� = �2 if � = � and � ≥ ,
�� = if � < � or � < .
• Equilibrium wage is � = � because the number of hired workers is
discontinuously increased even if the firm posts slightly higher wage than equilibrium wage.
Intuition: Diamond paradox
• In the case with positive search costs, the number of hired workers is
determined by two amount; (i) visiting workers, (ii) accepted workers given (i).
• Due to the information friction, the number of visiting workers is not affected by deviated wage.
• The number of accepted worker is affected, however, all workers are accepted e e if age is slightl lo e tha othe ’s age.
⇒ Goi g to the otto !!!!
Infinitely many firms
• The Diamond paradox hold even if the number of firms is many.
• Let focus on the case with infinite firms.
• The number of visiting firms is not affected by the deviation.
⇒A fi posts age as isiti g o ke s a ept it’s offe .
Infinitely many firms
• Value functions are
= + � + − ,
= − + � max� න
0
� I + − � �
• All firms post wages as
� = .
Infinitely many firms
• Values of employed and unemployed workers are, ഥ� = − � . and
= − − � . Therefore, the equilibrium wage is
ഥ� = − .
Robustness
• Even if workers can recall previous offers, Diamond paradox still holds.
• What’s happe i diffe e t elief fo ulatio .
I the u e t e sio , elief fo ulatio is passi e ← Deviation has no impacts on workers belief on the wage distribution.
↔ What’s happe if o su e s a k o the e a t dist i utio ???
← If the number of firms is enough large, Diamond paradox still holds.
(Intuition) The impact of an individual deviation on the wage distribution is almost zero.
Why paradoxical?
• Even in the exogenous value distribution, the equilibrium is not different in the perfect competitive model.
←However, both models are converging: If the search costs are very small, the sea h e uili iu is al ost sa e as i the o petiti e odel
⇔ In the endogenous value distribution, the search equilibrium is totally different with the competitive market model even if the search costs is almost zero.
Equilibrium wage are
• (almost) zero in the search model.
• Marginal revenue in the competitive market model.
Conclusion
• Diamond paradox shows the problem of simple price posting model. 1. No wage distribution.
2. Equilibrium wage is unrealistically too low. 3. Nobody search.
←Almost alternative (& tractable) models can solve some problem.
Diamond paradox
Diamond, P. A. (1971). A model of price adjustment. Journal of economic theory, 3(2), 156-168.
